A refinement of Vietoris' inequality for cosine polynomials

Let T-n(x) = Sigma(n)(k=0) b(k) cos(kx) with b(2k) = b(2k+ =1) = 1/4(k) ((2k)(k)) (k >= 0). In 1958, Vietoris proved that T-n(x) > 0 (n >= 1; x is an element of (0, pi)). We offer the following improvement of this result: The inequalities T-n(x) >= c(0) + c(1)x + c(2)x(2) > 0 (c(k) is an element of R, k = 0, 1, 2) hold for all n >= 1 and x is an element of (0, pi) if and only if c(0) = pi(2)c(2), c(1) = -2 pi c(2), 0 < c(2) <= alpha, where alpha = min(0 <= t<pi) T-6(t)/(t - pi)(2) = 0.12290....